On a Regularity Theorem for Weak Solutions to Transmission Problems with Internal Lipschitz Boundaries

Abstract
We show that if is a weak solution to <!-- MATH $\operatorname{div} (A\nabla u) = 0$ --> on an open set containing a Lipschitz domain , where <!-- MATH $A = kI{\chi _D} + I{\chi _{\Omega /D}}(k > 0,k \ne 1)$ --> 0,k \ne 1)$">. Then, the nontangential maximal function of the gradient of lies in <!-- MATH ${L^2}(\partial D)$ --> .